Analysis of circulating power within hybrid electric vehicle transmissions

Mechanism and Machine Theory 64 (2013) 131–143

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Analysis of circulating power within hybrid electric vehicle transmissions A.K. Gupta ⁎, C.P. Ramanarayanan Vehicles Research & Development Establishment, Ahmednagar (Maharastra) 414006, India

a r t i c l e

i n f o

Article history: Received 16 May 2012 Received in revised form 7 January 2013 Accepted 13 January 2013 Available online 1 March 2013 Keywords: Planetary gear drive Multiple paths Circulation of power Hybrid vehicle transmission

a b s t r a c t Since multiple paths of power flow exist in many transmissions, an undesirable situation of power circulation can arise in the system which leads to high mesh losses and less overall efficiency of the system. The paper presents a detailed analysis of circulating power within a planetary gear transmission used for hybrid vehicles. The fundamental relations connecting torques, speeds and power flows for a general planetary drive mechanism, are derived from the first principles for kinematic and power flow analysis. The power flow relations are investigated and clearly defined for the analysis of circulating power within complex chains of planetary drives and they also permit an immediate derivation of the power flow in all inversions of planetary drives. This study contributes to the development of a methodology for the circulating power-flow analysis of a planetary gear drive with two inputs and one output. This approach is generic and an illustration is presented to demonstrate its applicability to a hybrid planetary gear configuration. It is found that circulating power depends on planetary gear ratios and speed ratio of two power sources. © 2013 Elsevier Ltd. All rights reserved.

1. Introduction Many transmission systems have multiple paths of power transmission which can cause more losses associated with division of the input power(s) and allow the possibility of power recirculation in the system. In a planetary gear drive or differential gear train, the power circulating within the system elements may differ appreciably from that being transmitted, thus power recirculation or a power split configuration occurs. This phenomenon, which is a result of the relative motion of the gear members, is present in some degree in all planetary gear trains and has a significant effect on the performance and efficiency of systems transmitting power continuously [1]. The amount of circulating power may be either a small fraction of or several orders of magnitude greater than the input power, depending on the kinematic structure of planetary gear systems. If the concept of circulating power is not recognized, failure may occur in such systems. The circulating power inherent in the system has to be dealt with in order to minimize size and increase the overall efficiency of such transmissions. Therefore, in the design of transmission with planetary or differential gear sets, the issue of power recirculation is considered because power recirculation does not produce any useful output work [2]. A multi-mode hybrid electric vehicle transmission consists of one or more different types of a planetary gear train, which can cause the change in power flow pattern using mode-changing clutches to improve transmission efficiency. Due to increase in the environmental and economic interests, the potential of hybrid electric vehicles is expected to increase. Hybrid Electric vehicles (HEV) use different power sources with various powertrain configurations, including internal combustion engines, electric motors, electric generators, batteries and transmissions [3]. Among various alternative powertrains, the planetary gear hybrid powertrain is considered as one of the most promising configurations for hybrid electric vehicles [4]. Gear trains of hybrid vehicles such as Toyota Prius and Opel Astra can be recognized as planetary gear hybrid powertrains (PGHP).

⁎ Corresponding author. Tel.: +91 241 2544004; fax: +91 241 2548410. E-mail addresses: [email protected] (A.K. Gupta), [email protected] (C.P. Ramanarayanan). 0094-114X/$ – see front matter © 2013 Elsevier Ltd. All rights reserved. http://dx.doi.org/10.1016/j.mechmachtheory.2013.01.011

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A number of methods for determination of power flow and efficiencies of planetary gear drives have been proposed and expressions for a number of specific gear arrangements have been developed. Several methods are available for the analysis of torque and power flow for one DOF PGTs [5–8]. Macmillan [9] derived the power flow relations for the efficiency analysis of complex EGTs. Two procedures are presented to analyze the efficiency of any planetary gear train [10]. The first is based on a relationship between the output speed ratio and virtual gear teeth ratios, and in the second the efficiency can be written in terms of power transmitted through gear pairs and speed ratio. Chen and Angeles [11] demonstrated an approach based on the concept of virtual power to provide a power flow pattern through each element of any one-DOF EGT. Although many analytical expressions have been developed for determination of recirculation of power within the system, these relations are mostly for one input and one output drives. Kinematic analysis of a continuously variable transmission (CVT) utilizing non-circular gear sets and planetary summing differentials with an emphasis on the effect of different design parameters on reducing the circulating power in the system is performed [12]. The concept of virtual power is applied for static analysis and the efficiency of a two-DOFs epicyclic train, with associated applicable ranges [13]. GomàAyats, et al. [14] presented a methodology to analyze power transmitted through a mechanism comprising PGTs and branches with fixed and variable speed ratios starting from the kinematic analysis of the mechanism. An analytical process is developed for one input and one output coaxial epicyclic gear trains that divide input power and avoid internal recirculation. The process involves identifying power flows around internal closed loops and extracting for examination the arrangements that supply a required speed ratio [15]. A technique is demonstrated to describe the power flow by considering power flow through any single path of a general multi-path system. The method is illustrated with an application to a triple epicyclic train and the presence or otherwise of recirculated power within the system is investigated [16,17]. Ciobotaru, et al. [18] discussed an approach for analyzing multi-path power flow epicyclic transmissions. Recent analyses that address re-circulating power flow in planetary gear trains for hybrid electric vehicle are contained in [19–21]. Schulz [19] and Villeneuve [20] both discussed specific cases and investigated a specific power split transmission. Schulz derived the analytical conditions to avoid re-circulating power within main components of the Dual-E planetary gear hybrid powertrain [19]. It is a power-split hybrid electric drivetrain that can handle powers from an internal combustion engine (ICE) and two electric motors. Villeneuve [20] investigated a power split transmission developed by the Renault. Mattsson [21] derived a method to determine suitable values of basic speed ratios for a general CVT and investigated several power split transmissions. The graph based computation approach for kinematic and power flow analysis along with the mechanical efficiency analysis of epicyclic gear trains for parallel hybrid vehicles is discussed [22]. Planetary gear hybrid powertrains are analyzed while considering the circulating power within the system [23,24]. A systematic methodology for torque and power-flow analyses of multi-input multi-output epicyclic gear mechanisms (EGMs) with or without reaction link, based on the concept of fundamental circuit is developed [25]. This paper presents a method for analysis of circulating power in the planetary drives used for hybrid vehicles which is based on the fundamental relations derived from the first principles. The method offers a general method of solution, utilizing the kinematic characteristics of planetary systems. This method may be best illustrated by an example. A coupled planetary gear drive used in a parallel hybrid electric drive train with two inputs and one output is considered for an illustration. In this paper, the amount of circulating power incurred in the system is estimated by analyzing speed ratios and power transmitted by each component of the system. This method might be an extension of the analysis of re-circulating power methodology discussed in Ref. [15] to generic two-DOF planetary drives having two inputs and one output system. 2. Kinematic and power flow analysis A single planetary gear has three operating points: a sun gear (S), a carrier (C), and a ring gear (R) and two degrees of freedom as shown in Fig. 1(a). Fig. 1(b) shows a block diagram representation of a generalized two degree of freedom planetary gear system. A single degree of freedom is obtained by constraining one member so that motion given to one remaining member controls the motion of the other. Two planetary gear train combinations have five principle members and four degrees of freedom which requires three

2 R 1

3

S C

(a)

(b)

Fig. 1. (a) Planetary gear train schematic (b) block diagram representation.

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constraints for configuring one input-one output multispeed transmission system. These constraints may be as fixed members, without rotation, or inter-connections between members. The number of given input or output powers must be equal to degrees-of-freedom of the system. The configuration of the system depends on more than one input parameter in case of multi degrees of freedom systems. This implies that these mechanisms need multiple inputs in order to drive a unique output. A systematic approach is presented for the analysis of re-circulation of power within transmissions which include planetary gear hybrid vehicle drives. Power at connected members determines whether the input power(s) divides between internal paths or re-circulates within the system. The kinematics and power flow are governed by three fundamental Eqs. (1), (2) and (3) given below. For each simple planetary gear train (PGT) shown in Fig. 1, one can write the Willis' equation as follows: ω3 −Rω1 þ ðR−1 Þω2 ¼ 0 or, speed relation can be re-written as Rω1 þ ð1−RÞω2 ¼ ω3

ð1Þ

where R is the ratio of angular velocity ω3 to ω1, considering ω2 is equal to zero and defined as the basic velocity ratio of the planetary gear mechanism. Static equilibrium of the torques acting on shafts of the PG train requires that: th

T1 þ T2 þ T3 ¼ 0 with Ti being the torque of the i path:

ð2Þ

and T1 ω1 þ T2 ω2 þ T3 ω3 ¼ 0

ð3Þ

If all losses are negligible and direction of power flow is from member 1 to member 3, T1 T2 ¼ −T3 ¼ R1 ð1−R1 Þ

ð4Þ

Torques T1, T2 and T3 as well as angular velocities of member ω1, ω2 and ω3 all are defined as positive in the same direction. A positive power indicates that the power flows into the system and a negative power indicates that the flow is from the system. In other words, power is considered as positive when input to the mechanism and a negative power represents a power output of the mechanism. In general, for the two-input PGT without reaction member, any amount of power can be supplied to the train through any member whose torque and velocity directions are identical, i.e. both of them are either positive or negative. 2.1. Kinematic analysis of two planetary gear trains connected in parallel Two planetary gear trains can be connected in various manners. Consider the kinematics and power flow within the train as shown in Fig. 2, which consists of a planetary gear train of ratio R1 connected in parallel with another planetary gear train of ratio R2. The members of PGT1 and PGT2 are arbitrary numbered as 1, 2, 3 and 4, 5, 6 respectively. Interconnections of two planetary units with two inputs and one output, or their inversions with a reversed direction of power flow can always be represented in the form of Fig. 2. Thus if two power source arrangements are of interest with the transmission, all combinations of two planetary units are contained in this diagram. Input members are identified as 1 and 4 and it is assumed that their angular velocities are known in advance. Power is flowing through member 1 to 3 and member 4 to 6 in the PGT1 and PGT2 respectively, and the direction of power flow is considered positive as marked in Fig. 2. Any rigid connection between any two members makes them rotate at the same angular velocity. Thus, it can be observed, ω2 ¼ ω5 ; ω3 ¼ ω6 ¼ ωo ;

ð5Þ

P1 1

PGT1 (R1) 3

P2

P5

2

5

PGT2 (R2)

P4 4

6 P3

P6 PO= -(P3+ P6)

Fig. 2. Two planetary gear train (parallel-connected).

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and α is defined as the ratio of the angular velocity of two input power sources and is given by, ω4 ¼ α or ω4 ¼ αω1 : ω1

ð6Þ

Kinematic considerations for the planetary gear 1–2–3 and 4–5–6 show that (from Eq. (1)) R1 ω1 þ ð1−R1 Þω2 ¼ ω3

ð7Þ

R2 ω4 þ ð1−R2 Þω5 ¼ ω6

ð8Þ

where R1 is defined as the ratio of the output speed to input speed across members 3 and 1 when member 2 is held stationary, and similarly R2 is defined as the output to input speed ratio across members 6 and 4 when third member 5 is considered to be stationary. The overall speed ratio r1 between member 3 and member 1 is obtained by eliminating ω2, ω4, ω5 and ω6 from the expressions (5), (6), (7) and (8). On simplifying, the overall speed ratio r1 between member 3 and member 1 can be expressed in concise form as r1 ¼

ω3 R1 ð1−R2 Þ−α R2 ð1−R1 Þ : ¼ ω1 ðR1 −R2 Þ

ð9Þ

The overall speed ratio r2 between member 6 and 4 can also be written in terms of ω6 and ω4 of the planetary gear train 4–5–6 and is given by ω6 R ð1−R2 Þ−α R2 ð1−R1 Þ : ¼ r2 ¼ 1 αðR1 −R2 Þ ω4

ð10Þ

The expression for overall speed ratio implies that once the individual planetary gear ratios R1 and R2 are selected, the speed ratio is controlled by the factor α. The meaning of α is ratio of the angular speeds of one power source to another power source. This gives the theoretical base to control strategy design and simultaneously it becomes a constraint on the design of the overall planetary gear drive. This new approach can be specifically applied to planetary gear parallel transmissions used in hybrid electric vehicles. 2.2. Power-flow analysis of two planetary gear trains connected in parallel Consider the generalized block diagram of two PGTs connected in parallel again as shown in Fig. 2. Two input powers P1 and P4 from two different power sources are applied to member 1 of PGT 1 and member 4 of PGT 2 respectively. Due to possibility of the power flow in two paths, the power is divided in the parallel paths through two planetary gears. Power P1 or torque T1 is applied to PGT1 of planetary gear ratio R1. Probably only a percentage P2 of the input power P1 goes through the R2 planetary mechanism and remaining part (P1 − P2) goes to the output. Similarly, Power P4 or torque T4 at the PGT2 of planetary gear ratio R2 is imposed on two parallel paths. Out of the input power P4, only a part P5 passes through the R1 planetary and a difference of that (P4 − P5) bypasses R1 to add to the output of the transmission. The re-circulating power-flow analysis within the system is typically carried out by considering no power losses. In fact, usually these losses are not large enough to cause the change in the pattern of power-flow distribution significantly in planetary gear transmissions. Any part or the entire PGT must be under torque balance and power balance. If P1 and P4 are the input powers and Pi is power of the ith path, based on the torque and power equations we have following equations along with Eq. (4) for the planetary gear drive 1–2–3, T1 þ T2 þ T3 ¼ 0

ð11Þ

T1 ω1 þ T2 ω2 þ T3 ω3 ¼ 0:

ð12Þ

Similarly for the planetary gear drive 4–5–6, torque and power flow equations can be written as that for the planetary gear drive 1–2–3. T4 þ T5 þ T6 ¼ 0

ð13Þ

T4 ω4 þ T5 ω5 þ T6 ω6 ¼ 0

ð14Þ

T4 T5 ¼ −T6 ¼ R2 ð1−R2 Þ

ð15Þ

If the power has a positive algebraic sign, then the link is driving, otherwise is driven link. When the link is fixed, it cannot transmit any power i.e. power flow is zero. Expressions are derived for power flow ratios from relations (4), (11) and (12) in terms of the input power. Power flow ratios x1 and y1 are obtained in terms of the speed ratios and planetary gear ratio.

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Power flow ratio x1 is expressed as; x1 ¼ where

P2 ð1−R1 Þω2 =ω1 ¼ P1 R1

ð16Þ

ω2 1 ¼ ðr1 −R1 Þ from relation (7) ω1 1−R1

Substituting r1 from relation (9) and simplifying, x1 ¼

ðR1 −1ÞðR1 –αR2 Þ : R1 ðR1 –R2 Þ

ð17Þ

Power flow ratio y1 is expressed as; y1 ¼

P3 r1 ¼ : P1 R1

ð18Þ

On substitution of value of r1 from relation (9), y1 ¼

R1 ð1−R2 Þ−αR2 ð1−R1 Þ : R1 ðR1 −R2 Þ

ð19Þ

Similarly, power flow ratios x2 and y2 for the planetary gear drive 4–5–6 can also be obtained from Eqs. (13), (14) and (15), P5 ð1−R2 Þω5 =ω4 ¼ P4 R2 ω5 1 where (ω5/ω4) is given by ¼ ðr 2 −R2 Þ from relation (8), therefore ω4 1−R2 x2 ¼

x2 ¼

ðR2 −1ÞðR1 –αR2 Þ : αR2 ðR1 –R2 Þ

ð20Þ

ð21Þ

Power flow ratio y2 is obtained as; y2 ¼

P6 r ¼ 2 : P4 R2

ð22Þ

Further using relation (10), y2 can be expressed as y2 ¼

R1 ð1−R2 Þ−αR2 ð1−R1 Þ : αR2 ðR1 −R2 Þ

ð23Þ

If in any case, these relations are not satisfied, then the direction of power flow is reversed through the basic planetary unit. In the expressions (17), (19), (21) and (23) of power flow ratios, the power flow is such that P2, P3 and P5, P6 are power output of respective planetary gear drive. Thus, it is possible to express any of the ratios of the powers in terms of any single desired overall speed ratio, α and/or planetary gear ratios. 3. Analysis of circulating power Since the re-circulating power may exceed the transmitted power, this condition should be avoided in order to increase the overall mechanical efficiency. A case is reported where an excessive amount of re-circulating power caused the failure of a planetary gear train [26]. The amount of circulating power in the system depends upon the amount of power in the different components. In any hybrid transmission, two input powers are provided to the system i.e. both P1 and P4 are positive as shown in Fig. 2. Power flow direction within the system varies according to rotation of members and thus on the inter-connections between planetary members. It can be observed that only three conditions are possible in each planetary gear set for different direction of power flow within the system as mentioned in ref. [15]. For the planetary gear drive 1–2–3, power flow ratio x1 can have any one value of the following three constraints; 0bx1 b1 or x1 > 1 or x1 b0: Accordingly, direction of power flow for the three limits considered is shown in Table 1.

ð24Þ

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Table 1 Operational limits and power flow direction for the Planetary R1. Limits

Direction of power flow

0 b x1 b 1

x1 > 1 (Positive recirculation)

x1 b 0 (Negative recirculation)

Similarly, for the planetary gear 4–5–6, power flow ratio x2 can have any one value of the following three limits; 0bx2 b1 or x2 > 1 or x2 b0:

ð25Þ

In this case also, direction of power flow for the three limits considered is shown in Table 2. Limits of power flow ratios x1 and x2 are represented in the form of α, R1, R2 and overall speed ratios r1 & r2 in Table 3. Each limit of operation depends on the sign of power flow ratios. Limits of r1, r2, R1, and R2 for various values of α can be calculated from Table 3. It is understood that the circulation of power depends on the direction of power flow which in turn, depends upon sign of power flow ratios x1 and x2. The sign of power split ratio represents the direction of power delivery [27]. Direction of power flow in the members 2 and 5 plays an important role to provide the overall power flow pattern in two inputs and one output system. This depends upon the algebraic difference of x1 and x2 i.e. (x1 − x2) because of the two inputs to the system. Again, (x1 − x2) can have following three conditions: ) x1 −x2 b 0 or x1 −x2 > 0 or : x1 −x2 ¼ 0

ð26Þ

Any of the above three conditions along with conditions (24) and (25) decides the direction of overall power flow in the system. Two sources of power having the same power capacities are considered for derivation of the expressions and graphical representation of re-circulating power within the system in this paper. The effect of limits of (x1 − x2) along with the individual limits of x1 and x2 on the circulation of power in the system is studied. A total of nine possible cases of power flow in combination with the individual limits of x1 and x2 are analyzed and illustrated in Table 4. In case (i) as shown in Table 4, the power flow diagram shows that systems R1 and R2 have a characteristic in which power is always split at the whole speed ratio and is added to provide the output power of the transmission. A negative recirculation of power is occurred corresponding to case (ii) in the system, which is an undesirable condition. Considering the combination of limits of operation as in case (iii), the output power is provided at the output shaft without re-circulation of power in the system.

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Table 2 Operational limits and power flow direction for the Planetary R2. Limits

Direction of power flow

0 b x2 b 1

x2 > 1 (positive recirculation)

x2 b 0 (negative recirculation)

As can be seen from the power flow diagram described in case (iv), the power is split at the planetary gear units and the power is always circulated in the system. In all the combination of the three limits of (x1 − x2), power recirculation is occurred as shown in the power flow diagram of case (v) and can be considered as an undesirable condition. Since the power is re-circulating through the PGTs in case (vi), the input power instead of splitting power through the gear system is adding power to the re-circulating power. This is again an undesirable case because the power is recirculated within the system. In case (vii), power is split at the planetary gear units and a positive recirculation of power can be seen from the corresponding power flow direction. Compared to the power-split case, power recirculation changes the direction of flow in case (viii). An infeasible condition is appeared in the transmission mechanism in case (ix). The system efficiency is poor in the power recirculation situations and large sizes of power sources are

Table 3 Relations between operational limits and r1, r2, α, R1, R2. x1 and x2

r1, r2,R1, R2

α, R1, R2

0 b x1 b 1

ðr1 −R1 Þ b1 0b R1 ðr2 −R2 Þ 0b b1 R2 ðr1 −R1 Þ >1 R1 ðr2 −R2 Þ >1 R2 ðr1 −R1 Þ b0 R1 ðr2 −R2 Þ b0 R2

ðR1 −1ÞðR1 –αR2 Þ b1 R1 ðR1 –R2 Þ ðR2 −1ÞðR1 –αR2 Þ b1 0b αR2 ðR1 –R2 Þ ðR1 −1ÞðR1 –αR2 Þ >1 R1 ðR1 –R2 Þ ðR2 −1ÞðR1 –αR2 Þ >1 αR2 ðR1 –R2 Þ ðR1 −1ÞðR1 –αR2 Þ b0 R1 ðR1 –R2 Þ ðR2 −1ÞðR1 –αR2 Þ b0 αR2 ðR1 –R2 Þ

0 b x2 b 1 x1 > 1 x2 > 1 x1 b 0 x2 b 0

0b

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Table 4 Direction of overall power flow in the system. S. No.

Limits of x1 and x2

Limits of (x1 − x2)

(i)

0bx1 b1 and 0bx2 b1

x1 −x2 > 0

x1 −x2 b0

x1 −x2 ¼ 0

(ii)

0bx1 b1 and x2 > 1

x1 −x2 b0

(iii)

0 b x1 b 1 and x2 b 0

x1 − (−x2) > 0 i.e. x1 þ x2 > 0

(iv)

x1 > 1 and 0bx2 b1

x1 −x2 > 0

(v)

x1 > 1 and x2 > 1

x1 −x2 > 0 x1 −x2 b0 x1 −x2 ¼ 0

(vi)

x1 > 1 and x2 b0

x1 −ð−x2 Þ > 0 i.e. x1 þ x2 > 0

Direction of overall power flow

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Table 4 (continued) S. No.

Limits of x1 and x2

Limits of (x1 − x2)

(vii)

x1 b0 and 0bx2 b1

−ðx1 þ x2 Þb0

(viii)

x1 b0 and x2 > 1

−ðx1 þ x2 Þb0

(ix)

x1 b0 and x2 b0

x1 −x2 > 0 x1 −x2 b0 x1 −x2 ¼ 0

Direction of overall power flow

necessary. The true direction of power flow may or may not match to the defined direction shown in Table 4. These cases will have the estimated values either positive or negative corresponding to the direction. The variation of the circulating power within the working range of the planetary gear drive can be considered as proportional to (x1 − x2) in the present paper. The less power splits or circulates, the higher the potential for system efficiency [28]. The system efficiency decreases rapidly in a power recirculation system. The direction of power flow is analyzed in the gear system where the kinematic relationships for speed, torque and power are discussed earlier in this paper. An illustration of how the choice of basic ratio R1 affects (x1 − x2) and, thus overall direction of power flow in the system is shown in Fig. 3(a), (b), (c), (d), (e) and (f). Six different values of α (− 0.5, −0.1, 0.25, 0.5, 0.9 and 1.1) and four values of the overall speed ratio r1 (− 2, − 0.5, 0.75 and 1.5) are considered to analyze the behavior of (x1 − x2) with respect to basic planetary ratio R1. In all cases, the value of (x1 − x2) increases in both the direction i.e. power is circulated more as magnitude of the planetary gear ratio R1 for the system decreases. This can be seen from Fig. 3(a) to (f). The value of (x1 − x2) is less for α = 0.9 and 1.1 when compared to all other considered cases and goes to maximum about 250 for α = −0.1 and r1 = 1.5 as illustrated in Fig. 3(e), 3(f) and Fig. 3(b) and (b) respectively. The latter case is a typical and highly undesirable where an excessive amount of power is re-circulated through the system and the power flowing through the gear system is much higher than the input powers. The value of (x1 − x2) is decreasing in positive as well as negative direction as magnitude of R1 increases in positive as well as negative side. For r1 = − 0.5, (x1 − x2) is found to be minimum in all the cases. However, the true direction of power flow would also depend on magnitude and sign of the power flow ratios y1 and y2 in addition to (x1 − x2). When the speed of two power sources are selected such that α = 1, overall speed ratios r1 and r2 become one for all values of R1 and R2. However, the power will flow according to values and sign of y1, y2 and (x1 − x2) which depends on R1 and R2. It is evident that α plays a critical role for the power flows in the planetary drives with two inputs and one output system. Further, for no recirculation of power any of following three conditions must satisfy:

0 b x1 b 1 and 0 b x2 b 1 for all values of ðx1 –x2 Þ 0 b x1 b 1; x2 b 0 and x1 −x2 > 0 with proper algebraic sign of x1 and x2 x1 b 0; 0 b x2 b 1 and x1 −x2 b 0 with proper algebraic sign of x1 and x2

) ð27Þ

4. Example of application An illustrative case for the kinematic and power relationship is given in this section. Fig. 4 shows a schematic diagram of a coupled planetary gear drive for hybrid vehicle applications. In practice, variations of the design are possible by selecting different combinations of various system parameters, depending on applications. Two input motions from the two power sources are delivered to the coupled planetary gear set. The resultant output motion is algebraically summed and delivered to the output shaft. Only one internal combustion engine (ICE) and one electric motor/generator (EM) are required for this configuration. This type of transmission mechanism has been installed in a prototype of parallel hybrid electric rear-wheel drive vehicle and tested.

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Fig. 3. Basic planetary ratio R1 and (x1 − x2) for various values of α and overall speed ratio r1.

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4

141

8

2

7

5 1

3

6

Fig. 4. Coupled planetary gear train as hybrid transmission.

Five basic modes of operation viz. motor-only, combined power, engine-only, engine/charge and regenerative braking modes are possible in this arrangement of the two planetary gear sets. An engine and an electric motor are connected to sun gear 1 and planetary carrier 7 respectively. It is apparent that the reaction member (in the present case, coupled ring gears 4 and 8) is used in two-input PGTs to control the direction of rotation of other members and thus the direction of the power flow. The planetary carrier 3 engages simultaneously with sun gear 5, which is keyed to the output shaft. The input powers are coming from two directions, sun gear 1 and planetary carrier 7. The input power that comes from the engine through the sun gear 1 split, a percentage goes through the planetary carrier 3 and the other part goes to the ring gear 4. Since ring gears 4 and 8 are coupled, this power goes to the sun gear 5 and planetary carrier 7. Thus the power goes to the output shaft passing through the sun gear 5. This power is added to the one coming from the planetary carrier 3 flowing through the planet gear, which transmits the total power to the output shaft. The input power that comes from the electric motor through the planetary carrier 7 split, a percentage goes through the sun gear 5 and the other part goes to the ring gear 8. The power from ring gear 8 coupled with the ring gear 4, goes to planetary gear 3 and sun gear 1. Since the planetary carrier 3 is coupled with the sun gear 5, this power is added to the power flowing through the sun gear 5, which in turn is delivered to the output shaft. In this way, the power circulates in the system. Since there is a power re-circulation and if small amount of the input powers passes through the ring gears, this makes it more efficient. The planetary carrier 3 and sun gear 5 collects the power flowing in the gear train and delivers the total output power. Following are constraint equations for the coupled planetary gear set: ω3 ¼ ω5 ; ω4 ¼ ω8 :

ð28Þ

Rewriting Eq. (9) of overall speed ratio r1 between member 3 and member 1, r1 ¼

ω3 R1 ð1−R2 Þ– α R2 ð1−R1 Þ : ¼ ðR1 −R2 Þ ω1

ð29Þ

7 Here, α is defined as ω ω1 ¼ α or ω7 ¼ αω1 . Substituting the values of R1 and R2 for this configuration in the Eq. (29), and with the help of Eqs. (7) and (8), output speed of the transmission is obtained as

ω3 ¼ ω7 þ

R2 ðω1 −ω7 Þ : R 1 ð1 þ R 2 Þ þ R 2

ð30Þ

On simplifying, ω3 ¼ ω7 þ

ω1 ð1−αÞ : R1 ð1 þ 1=R2 Þ þ 1

ð31Þ

This relation can also be expressed in terms of number of teeth on the sun and ring gears as ω3 ¼ ω7 þ

ω1 ð1−αÞ z41 ð1 þ z58 Þ þ 1

where zij denotes the ratio of number of teeth on gear i to gear j.

ð32Þ

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A.K. Gupta, C.P. Ramanarayanan / Mechanism and Machine Theory 64 (2013) 131–143

Now, power at each component is expressed in terms of the input power with an assumption of 100% overall efficiency of the gear systems. Power flow ratios for this configuration using Eqs. (17), (19), (21) and (23) and by replacing the planetary ratios with number of teeth of respective gears, can be found as x1 ¼

P4 z41 ðαð1 þ z41 Þð1 þ z85 Þ−1Þ ¼ 1–ð1 þ z41 Þð1 þ z85 Þ P1

ð33Þ

y1 ¼

P3 z85 ð1 þ z41 Þ þ αz41 ð1 þ z41 Þð1 þ z85 Þ ¼ P1 ð1 þ z41 Þð1 þ z85 Þ−1

ð34Þ

x2 ¼

P8 z85 ð1−αð1 þ z41 Þð1 þ z85 ÞÞ ¼ P7 α ðð1 þ z85 ÞÞð1–ð1 þ z41 Þð1 þ z85 ÞÞ

ð35Þ

y2 ¼

P5 z85 þ αz41 ð1 þ z85 Þ : ¼ P7 α ðð1 þ z85 ÞÞðð1 þ z41 Þð1 þ z85 Þ−1Þ

ð36Þ

Depending upon the number of teeth on the sun gears, ring gears and value of α, power flow ratios x1 and y1, x2 and y2 can be determined. Thus the circulation of power in the hybrid transmission can be known. This further helps in designing the transmission. The validity of results requires that the directions of power flow remain the same even in the presence of all losses, as of those computed with 100% efficiency. This condition is assumed to be satisfied while deriving the equations. 5. Conclusion In this study, a general formulation for kinematic and power flow analysis emphasizing on re-circulating power in the two inputs and one output planetary gear systems is developed in a systematic manner. The speed and power flow of various planetary gear trains of this type can be expressed by the basic formulas developed in the study. These relations can be used to understand power flow within complex gear trains. By enabling to control the flow, a desired overall kinematics can be achieved while circulation of power in the system is held to a minimum. Two planetary gear combination units and all its inversions can be analyzed by the method discussed herein the paper. Overall speed ratio and power flow ratios depend mainly on three parameters; α (speed ratio of two power sources), planetary ratios R1 and R2. The magnitude of R in these expressions is a function of the type of planetary train and gear size. This method is perhaps the extension of the previous methods provided for re-circulating power in the transmission with one input and one output system. This approach is generic and an illustration is presented to demonstrate its applicability to a coupled planetary gear transmission configuration that has practical significance in automotive applications. This gear train is used in a parallel hybrid electric vehicle since it accepts two inputs and delivers one output. References [1] E.I. Radzimovsky, A simplified approach for determining power losses and efficiency of planetary gear drives, Machine Design 28 (3) (1956) 101–110. [2] V.H. Mucino, Z. Lu, J.E. Smith, M. Kimcikiewicz, B. Cowan, Design of continuously variable power split transmission systems for automotive applications, Proceedings of the Institution of Mechanical Engineers, Part D: Journal of Automobile Engineering 215 (2001) 469–478. [3] Matthew A. Kromer, and John B. Heywood, A comparative assessment of electric propulsion systems in the 2030 US light duty vehicle fleet, SAE paper No. (2008) 2008-01-0459. [4] Sungtae Cho, Kukhyun Ahn, Jang Moo Lee, Efficiency of the planetary gear hybrid powertrain, Proceedings of the IMechE, Part B: Journal of Engineering 220 (2006) 1445–1454. [5] E. Pennestri, F. Freudenstein, The mechanical efficiency of epi-cyclic gear trains, ASME Journal of Mechanical Design 115 (1993) 645–651. [6] F. Freudenstein, A.T. Yang, Kinematics and statics of a coupled epicyclic spur-gear train, Mechanism and Machine Theory 7 (1972) 263–275. [7] E. Pennestri, F. Freudenstein, A systematic approach to power-flow and static force analysis in epicyclic spur-gear trains, ASME Journal of Mechanical Design 115 (3) (1993) 639–644. [8] H.I. Hsieh, L.W. Tsai, Kinematic analysis of epicyclic-type transmission mechanisms using the concept of fundamental geared entities, ASME Journal of Mechanical Design 118 (1996) 294–299. [9] R.H. Macmillan, Power flow and loss in differential mechanisms, Journal of Mechanical Engineering Science 3 (1) (1961) 37–41. [10] J.M. del Castillo, The analytical expression of the efficiency of planetary gear trains, Mechanism and Machine Theory 37 (2002) 197–214. [11] C. Chen, J. Angeles, Virtual-power flow and mechanical gear-mesh power losses of epicyclic gear trains, ASME Journal of Mechanical Design 129 (2007) 107–113. [12] D. Dooner, H-D Yoon, A. Seireg, Kinematic considerations for reducing the circulating power effects in gear-type continuously variable transmissions, Proceedings of the Institution of Mechanical Engineers, Part D: Journal of Automobile Engineering 212 (D6) (1998) 463–478. [13] C. Chen, T.T. Liang, Theoretic study of efficiency of two-DOFs of epicyclic gear transmission via virtual power, ASME Journal of Mechanical Design 133 (2011) 031007-1–031007-7. [14] J.R. GomàAyats, et al., Power transmitted through a particular branch in mechanisms comprising planetary gear trains and other fixed or variable transmissions, Mechanism and Machine Theory 46 (2011) 1744–1754. [15] G. White, Derivation of high efficiency two-stage epicyclic gear, Mechanism and Machine Theory 38 (2003) 149–159. [16] D.J. Sanger, The determination of power flow in multiple — path transmission systems, Mechanism and Machine Theory 7 (1) (1972) 103–109. [17] Wojnarowski, Comments on ‘The determination of power flow in multiple — path transmission systems’, Mechanism and Machine Theory 10 (1975) 261–266. [18] T. Ciobotaru, et al., Method for analyzing multi-path power flow transmissions, Proceedings of the IMechE, Part B: Journal of Engineering Manufacture 224 (2009) 1447–1454. [19] M. Schulz, Circulating mechanical power in a power-split hybrid electric vehicle transmission, Proceedings of the Institution of Mechanical Engineers, Part D: Journal of Automobile Engineering 218 (2004) 1419–1425. [20] A. Villeneuve, Dual mode electric infinitely variable transmission, SAE 04CVT-19, 2004.

A.K. Gupta, C.P. Ramanarayanan / Mechanism and Machine Theory 64 (2013) 131–143

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[21] P. Mattsson, Continuously Variable Split-power Transmission with Several Modes, Chalmers University of Technology, Goteborg, Sweden, 1996. [22] E. Pennestrì, et al., Efficiency evaluation of gearboxes for parallel hybrid vehicles: theory and applications, Mechanism and Machine Theory 49 (2012) 157–176. [23] H. Yang, et al., Analysis of planetary gear hybrid powertrain system part 1: input split system, International Journal of Automotive Technology 8 (6) (2007) 771–780. [24] H. Yang, et al., Analysis of planetary gear hybrid powertrain system part 2: output split system, International Journal of Automotive Technology 10 (3) (2009) 381–390. [25] Essam L. Esmail, Shaker S. Hassan, An approach to power-flow and static force analysis in multi-input multi-output epicyclic-type transmission trains, Transactions of the ASME Journal of Mechanical Design 132/011009-1-011009-10 (2010). [26] E. Buckingham, Spur Gears, McGraw-Hill Book Co., 1928. [27] B. Pohl, CVT split power transmission, a configuration versus performance study with an emphasis on the hydromechanical type, SAE paper No. (2002) 2002-01-0589. [28] D. Fussner, and Y. Singh, Development of single stage input coupled split power transmission arrangements and their characteristics. SAE paper No. 2002-01-1294.